Defects, nested instantons and comet-shaped quivers

We introduce and study a surface defect in four-dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a $$\mathrm{{D3/D7}}$$ D 3 / D 7 -branes system on a non-compact Calabi–Yau threefold X. For $$X=T^2\times T^*{{\mathcal {C}}}_{g,k}$$ X = T 2 × T ∗ C g , k , the product of a two torus $$T^2$$ T 2 times the cotangent bundle over a Riemann surface $${{\mathcal {C}}}_{g,k}$$ C g , k with marked points, we propose an effective theory in the limit of small volume of $${\mathcal C}_{g,k}$$ C g , k given as a comet-shaped quiver gauge theory on $$T^2$$ T 2 , the tail of the comet being made of a flag quiver for each marked point and the head describing the degrees of freedom due to the genus g. Mathematically, we obtain for a single $$\mathrm{{D7}}$$ D 7 -brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials naturally arises.